I'm hoping to find a reasonable value to assign to the divergent series $\sum_{n=0}^\infty (-1)^n n^n$ and $\sum_{n=0}^\infty (-1)^n (xn)^n$. For the first one, I have obtained something around 0.71, but I'm very unsure if this is correct. For the second series, I get a graph that looks about like this: 
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What sort of methods are powerful enough to sum this series? I think Borel summation is too weak, since this grows faster than any $\left(\alpha n\right)!$.
Any help would be appreciated!
EDIT: I obtained my values from a series of questionably valid approximations, but here is another method that seems to give a similar value.
The idea is that illegally swapping summation signs, and then continuing the inner summation past its regular convergent range allows one to assign a value to a divergent series.
Starting with
$$\sum_{n=0}^\infty (-1)^n n^n$$
We want to get two summations in order to be able to swap, so we expand into
$$\sum_{n=0}^\infty (-1)^n e^{n\ln(n)} = \sum_{n=0}^\infty (-1)^n \sum_{k=0}^\infty \frac{\left(n \ln(n)\right)^k}{k!}$$
Simplifying a bit more, we get
$$\sum_{n=0}^\infty (-1)^n \sum_{k=0}^\infty \frac{n^k \ln(n)^k}{k!} =\sum_{n=0}^\infty (-1)^n \sum_{k=0}^\infty \frac{e^{\ln(n)k} \ln(n)^k}{k!}$$
Swapping the summations, we get
$$\sum_{k=0}^\infty \frac{1}{k!} \sum_{n=0}^\infty (-1)^n e^{\ln(n)k} \ln(n)^k = \sum_{k=0}^\infty \frac{1}{k!} \sum_{n=0}^\infty \frac{d^k}{dk^k} (-1)^n n^k $$
(Note: I'm using $\frac{d^k}{dk^k}$ to represent taking the derivative with respect to k, k times).
Now, we can continue the inner summation by doing
$$\sum_{k=0}^\infty \frac{1}{k!} \frac{d^k}{dk^k}\sum_{n=0}^\infty (-1)^n n^k = \sum_{k=0}^\infty \frac{1}{k!} \frac{d^k}{dk^k}(1-\eta(-k))$$
To finally get
$$1/2 – (\eta'(-1) – \frac{\eta''(-2)}{2!} + \frac{\eta'''(-3)}{3!} – \dots)$$
The sum of the first few terms seems to be around .71, which agrees fairly closely with the other method.
EDIT2:
It looks like most methods converge onto about the same shape eventually. 
The green line is the asymptotic expansion, meaning I only took the first 5 terms in the power series, and left the rest out. Usually, this asymptotic expansion converges to the right function within a very small radius. The black function is using something like FusRoDah's method. The orange graph is using the non-rigourous series of approximations.
This next pictures shows the functions over larger intervals–the purple function is using that eta method I outlined above.
To elaborate more about how I used FusRoDah's method, I started with $BA(t)=\sum_{k=0}^\infty \frac {(-kt)^k} {k!} = 1+\sum_{k=1}^\infty \frac {(-kt)^k} {k!}$. Then, I added in the approximation to get
$$\operatorname{BA}(t) = 1+\sum_{k=1}^\infty \frac {(-kt)^k} {k!} + \frac {(-et)^k} {\sqrt{2\pi k}}-\frac {(-et)^k} {\sqrt{2\pi k}} = 1+\sum_{k=1}^\infty\frac {(-et)^k} {\sqrt{2\pi k}}+ \sum_{k=1}^\infty \frac {(-kt)^k} {k!} -\frac {(-et)^k} {\sqrt{2\pi k}}.$$
The first part of the sum can be written as an integral of the polylogirthmn with some other terms, but we are still left with the $\int_{0}^{\infty}e^{-t}\sum_{k=1}^{\infty}\left(\frac{\left(-ktz\right)^{k}}{k!}-\frac{\left(-etz\right)^{k}}{\sqrt{2\pi k}}\right)dt$ term. Since this doesn't converge on its own, we can approximate it with $$\int_{0}^{A}e^{-t}\sum_{k=1}^{B}\left(\frac{\left(-ktz\right)^{k}}{k!}-\frac{\left(-etz\right)^{k}}{\sqrt{2\pi k}}\right)dt.$$
Increasing $A$ and $B$ increases the accuracy, but decreases the range of convergence. I used $A = 2.5$, $B = 30$ for the first graph.
What remains to be done then is either find a more accurate approxation, such that $\sum_k \frac{(-kt)^k}{k!} – f_t(k)$ converges for all values of t, or to find a way to continue the difference between the approximation and the original series.
Best Answer
The sum is equal to $$ \int\limits_{0}^{\infty}\frac{\exp(-x)}{1+W_0(x)}\,\mathrm{d}x = 0.7041699604... $$ where $W_0(x)$ is the Lambert-$W$ function.
Reference
Stephen Finch. "Errata and Addenda to Mathematical Constants [Math.HO]", §6.11. Iterated Exponential Constants, pg. 66.
Two interesting divergent sums from this reference are as well $$\sum_{n=1}^\infty (-1)^{n-1} (2n)^{2n-1} = \int\limits_{0}^{\infty}e^{-x}\log(x/|W_{0}(i x)|)\mathrm{d}x = 0.3233674316...$$ which seems to be also the cosine and sine integrals of the derivatives $W'_{0}(x) = \frac {W_{0}(x)}{x(1 + W_{0}(x))}$ and $-W''_{0}(x)$ respectively $$\sum_{n=1}^\infty (-1)^{n-1} (2n)^{2n-1} = \int\limits_{0}^{\infty}\mathrm{cos}(x) W'_{0}(x)\mathrm{d}x = -\int\limits_{0}^{\infty}\mathrm{sin}(x) W''_{0}(x)\mathrm{d}x$$ as it can be seen in the following Pari GP v2.14.0 lines
although, as it is pointed out, a rigorous proof is not yet known.
And this variation, $$\sum_{n=1}^\infty (-1)^{n-1} (2n-1)^{2n} = \frac{1}{2i}\int\limits_{0}^{\infty}e^{-x}(g(ix)-g(-ix))\mathrm{d}x = 0.0111203007...$$ where $g(x) = W_{0}(x)/(1 + W_{0}(x))^3$